Let be a matroid defined on set . Let .
Two matroids and are said to be isomorphic if there is a bijection between their underlying sets and that preserves independence.

Operations on Matroids

The Matroid contraction of to is denoted as is a matroid whose cycles are the minimal members of the collection where is a cycle of .
The Matroid restriction of to is denoted as is a matroid whose cycles are those contained in .
A Minor is a matroid obtained either by restriction or construction
- (Wilson e31.9.1) For , then
  - Let be a cycle of Performing gives a matroid with the minimal members of as a cycle.
    
    Since , . Therefore, the above is equivalent to and the corresponding matroid is .
- (Wilson e31.9.2) For , then
  - Proof: Performing gives us a new matroid whose cycles are contained in .
    
    Since , any cycle contained in
    
    So we could get the same result if we performed instead.
Let be matroids on the same set . Then, a new matroid called their union, denoted can be defined by taking as independent set, all possible unions of an independent set in .
- (Wilson 33.3) Let be rank functions to their corresponding matroids. Then, the rank function of the union is given by

Theorems

(Wilson e31.10.1) Any minor of a graphic matroid is also graphic.
- Proof: Since matroid contraction and matroid restriction correspond to edge contraction and edge deletion we have that the resulting matroid minor is still graphic as it corresponds to a new graph with some edges removed or contracted, and no new edges were added so no new spanning forests were added.
(Wilson e31.10.2) Any minor of a cographic matroid is also cographic.
- Proof: Similar to (Wilson e31.10.1), except applying to cotrees.
(Wilson e31.10.3) Any minor of a regular matroid is also regular.
- Proof: If is regular, then it is representable in every field . So, there exists a map and a Vector Space that maps independent sets of to linearly independent subsets of .
  
  Performing either matroid restriction or matroid contraction is equivalent to considering a subset of a set linearly independent vectors. This subset is also linearly independent so the resulting matroid is also regular
(Wilson e31.10.4) Any minor of a binary matroid is also binary.
- Proof: Similar to (Wilson e31.10.3) except restricted to the field of .